Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909674822959104 |
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| author | Gallese, Andrea Goodson, Heidi Lombardo, Davide |
| author_facet | Gallese, Andrea Goodson, Heidi Lombardo, Davide |
| contents | Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. In this paper, we compute the Sato-Tate group of $J_m$. Currently, there is no general algorithm that computes this invariant. We also describe the Sato-Tate group of an abelian variety, generalizing existing results that apply only to non-degenerate varieties, and prove an extension of a well-known formula of Gross-Koblitz that relates values of the classical and $p$-adic gamma functions at rational arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups Gallese, Andrea Goodson, Heidi Lombardo, Davide Number Theory Algebraic Geometry 11F80, 11G10, 14C25, 11G15, 14K15 Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. In this paper, we compute the Sato-Tate group of $J_m$. Currently, there is no general algorithm that computes this invariant. We also describe the Sato-Tate group of an abelian variety, generalizing existing results that apply only to non-degenerate varieties, and prove an extension of a well-known formula of Gross-Koblitz that relates values of the classical and $p$-adic gamma functions at rational arguments. |
| title | Monodromy groups and exceptional Hodge classes, II: Sato-Tate groups |
| topic | Number Theory Algebraic Geometry 11F80, 11G10, 14C25, 11G15, 14K15 |
| url | https://arxiv.org/abs/2507.02535 |